Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C25 |
| direct product | C5×#1869 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C23 |
| commutator | C22 | C22×C5×C8 |
| Frattini, cosocle | C22×C4 | C23×C5 |
| socle | C23×C5 | C22×C4 |
| prime radical | C22×C5×D4 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ G | |
432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,3,4,5)(7,9)(10,11)(12,13)(14,15,17,20,18,16,19,21);(6,7,8,9)(14,16,17,21,18,15,19,20);(11,13)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 5459);